An exact period criterion for focal-distance products in elliptic billiards

For a noncircular elliptic billiard with a strictly nested nondegenerate confocal elliptical caustic, the product of ordinary distances from all vertices to either fixed focus is independent of phase if and only if the primitive least period is congruent to 2 modulo 4. The result covers all admissible primitive star turns and both orientations. For every even least period, the two focal products are equal at each phase. Repetitions raise the positive primitive product to an integer power, so listed return length alone is insufficient. Circles are treated separately; hyperbolic and degenerate caustics are excluded. This research note proves the primitive interpretation of the experimentally recorded invariant k114 in Reznik–Garcia–Koiller (2020/2021), establishes the exact period classification, and gives an exact doubled-triangle counterexample to the unrestricted six-event return interpretation. It applies Stachel's published parametrization, classical Jacobi identities and established meromorphic divisor methods. No new-method or exclusive historical priority claim is made. The bounded primary-source audit found no explicit earlier proof of this exact target; novelty of the specific formulation is not certified. Full text of one related 2023 Dragović–Gajić article was unavailable, and the accompanying priority note records that coverage gap. The deposit contains the manuscript PDF and a support archive with standalone LaTeX source, exact finite arithmetic controls, independent supplementary theta-series diagnostics, recorded outputs, metadata, references and priority-search limits. Finite checks support the analytic all-period proof; floating-point computations are not validated numerical certification. AI tools were used extensively in solving, drafting, source comparison, reproduction and independent adversarial reviews. This preprint is unrefereed and has not undergone conventional human peer review. The author is responsible for its claims.